# Lesson 23 Expected Value and Infinity

## 23.1 Pascal’s Wager

The mathematician Blaise Pascal (1623-1662) made an argument for why it was rational to believe in God—at least the Judeo-Christian God. Pascal acknowledged that God may or may not exist and assigned the probability \(p > 0\) to God’s existence. Then, he considered the following bets:

- If you believe in God, and He exists, then your reward is infinite (i.e., eternal salvation). If He does not exist, then your reward is some number \(a\), which may be negative but finite.
- If you do not believe in God, and He exists, the your reward is negative infinity (i.e., eternal damnation). If He does not exist, then your reward is some number \(b\), which is finite.

Looking at the first bet (belief in God), Pascal calculated the expected reward \(E[R]\) as follows. The p.m.f. of \(R\) is \[ \begin{array}{r|cc} r & a & \infty \\ \hline f(r) & 1-p & p, \end{array} \] and the expected value of this bet is \[ E[R] = a\cdot (1-p) + \infty \cdot p = \infty. \]

Can you do the same for the second bet?

Pascal argued that we should pick the bet with the higher expected value, which is to believe in God.

I present this only as an illustration of how expected value can be applied in surprising contexts, not to convince you to believe or not to believe. There have been many philosophical objections to Pascal’s wager. If you are interested, this video discusses some of the debates around Pascal’s wager. You can find many more discussions on Youtube.

## 23.2 St. Petersburg Paradox

In the Pascal’s Wager example, we dealt with random variables which could take on values such as \(\infty\) or \(-\infty\). It is not surprising that the expected value is infinite when infinity is a possible value. However, the expected value can be infinite, even if the random variable is finite-valued. Let’s look at an example.

You are offered the following game at a carnival. The game starts with $1 in a jar. You toss a coin repeatedly. Each time the coin lands heads, the amount of money in the jar is doubled. As soon as the coin lands tails, you cash out whatever money is in the jar, and the game ends.

For example, if you toss the sequence HHHT, then you win $8 because:

- After the first H, the amount in the jar is doubled to $2.
- After the second H, the amount in the jar is doubled to $4.
- After the third H, the amount in the jar is doubled to $8.
- After the T, the game ends, and you win the $8 in the jar.

How much would you be willing to pay to play this game?

We have seen that the expected value represents the “fair value” of a game. First, let’s work out the p.m.f. of the amount of money you win, \(W\). Fill in the first few probabilities of this p.m.f. table. \[ \begin{array}{rcccccc} w & 1 & 2 & 4 & ? & ? & \ldots \\ \hline f(w) & 1/2 & 1/4 & ? & ? & ? & \ldots \\ \end{array}\]

Now calculate \(E[W]\) from the p.m.f. What does this suggest about how much
you should be willing to pay? This is a famous puzzle in probability called the
**St. Petersburg Paradox**.

Here is a video discussing the St. Petersburg paradox.